Analyticity and Integrality

Analyticity and the Complex Logarithm

We consider a complex-valued function $f$ of one complex variable $z$, and we write:
\[ \int_{\mathrm{A}} f(z) d z=(\mathrm{A}, f(z) d z)\]
interpreting it as being bilinear in the differential \( f(z) d z \) and the homological $1$-chain \( \mathrm {A} \).

Then, if \( f \) is a continuous function of the complex variable \( z \) in the region $\mathbf R$ and if it holds:
\[\int_{\mathrm{C}} f(z) d z=0\]
about any closed chain $\mathrm {C}$ in \( \mathbf{R}, \) then \( f(z) \) represents an analytic function in $\mathbf {R}$.

In practice, if \( f \) is analytic merely on a closed path \( \mathcal{C}, \) it does not follow that \( \int_{\mathrm{C}} f(z) d z=0 . \) In particular,
\[\int_{\mathrm{C}} \frac{d z}{z}=2 \pi i\]
where \( \mathcal{C} \) is a simple closed curve embracing the origin.

This leads to the conclusion that it is precisely the complex logarithm function that characterizes the geometry of the complex plane in all its essential aspects.

The integrand \( d z / z \) is written as \( d \log z \) and we obtain:
$$\begin{aligned} \int_{\mathrm{C}} \frac{d z}{z}=\int_{z \text { on } \mathrm{C}} d \log z=\left.\log z\right|_{\mathrm{C}}=\log |z|+\left . i \arg z\right|_{\mathrm{C}}=2 \pi i \end{aligned}$$

meaning that if \( z \) is parametrized as $z(t)=x(t)+i y(t)$ by a continuous variation of $t$ from $0$ to $l$,
$$\left.\log z\right|_{\mathrm{C}} =\log z(l) - \log z(0)=2\pi i $$
Interestingly enough, \( d \log z \) is always single-valued as a differential despite the fact that the indeterminacy of log \( z \) can be any number of the type \( 2 \pi i \kappa \) for \( \kappa \) a positive or negative integer.

Comments

Post a Comment

Popular posts from this blog

Polar Spider Web

The Spider Web Exponential Epimorphism  The exponential homomorphism \[\exp : \mathbf{C} \rightarrow \mathbf{C}^{\mathbf{x}}, \quad \boldsymbol{z} \mapsto \exp z \] is clearly an epimorphism, that is, surjective, but it is not injective. From the complex exponential group epimorphism and the fact that \( \operatorname{\mid exp} z \mid=1 \text { holds if and only if } z \in \mathbb{R} i \) it can be deduced that \( \exp (i \mathbb{R})=S^{1}, \) where \( S^{1} \) denotes the multiplicative circle group. Hence, we obtain the following  polar spider web epimorphism: The morphism \( p: \mathbb{R} \rightarrow S^{1}, \varphi \mapsto e^{i \varphi} \) is a group epimorphism whose kernel is the group \( 2 \pi \mathbb{Z}, \) and we have: \( p\left(\frac{\pi}{2}\right)=i .\) Note that for \( \varphi, \psi \in \mathbb{R} \) we have: \( p(\varphi+\psi)=\exp (i \varphi+i \psi)=(\exp i \varphi)(\exp i \psi)= \) \( p(\varphi) p(\psi) . \) Thus, \( p \) is an epimorphism, since \( p(\mathbb{R...
Read more

Integral Cohomology and Geometry

Geometric Realization of Second Integral Cohomology Classes The second integral cohomology classes in \( H^{2}(M ; Z) \) are in natural \( 1-1 \) correspondance with the isomorphism classes of circle-bundles over \( M \), where $M$ is a manifold. In particular, every second integral cohomology class \( \hat{F} \in H^{2}(M ; Z) \) has a geometric realization which consists of a space \( P=P_{F} \) together with an action of the circle \( S^{1} \) on \(P \) and a natural projection: \[\begin{array} \\ P \\ \downarrow \pi \\ M \end{array}\] so that the \( S^{1} \) action on \( P \) is free, preserves \( \pi, \) and \( \pi \) induces an isomorphism of \( P / S^{1} \) with \( M .\) For instance if \( M \) is a Riemann surface, consider that $P$ is the set of its unit tangent vectors to $M$, and let \( \pi \) be the projection map which assigns to each tangent vector the corresponding point of tangency. The circle \( S^{1} \) then acts freely by rotating any vector counter-clockwise in its...
Read more